A uniqueness theorem for harmonic functions on the upper-half plane

Abstract

Consider harmonic functions on the upper-half plane R + 2 = { ( x , y ) | y > 0 } R^{2}_{+}\! = \{(x,y)|\;y > 0 \} satisfying the boundary condition u y = − exp ⁡ ( u ) u_{y}=-\exp (u) and the constraint ∫ R + 2 exp ⁡ ( 2 u ) > ∞ \int _{R^{2}_{+}}\exp (2u) > \infty . We prove that all such functions are of form (1.2) below.